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Chapter 4: Problem 20

Solve the equation. \(27^{x-4}=9^{2 x+1}\)

Short Answer

Expert verified
x = -14

Step by step solution

01

Express bases as powers of 3

Rewrite the given equation by expressing 27 and 9 as powers of 3. Note that 27 = 3^3 and 9 = 3^2. So, the equation becomes: 3^{3(x-4)} = 3^{2(2x+1)}.
02

Simplify the exponents

Now, simplify the exponents in the equation:3^{3x-12} = 3^{4x+2}.
03

Set the exponents equal to each other

Since the bases are the same, we can set the exponents equal to each other:3x - 12 = 4x + 2.
04

Solve for x

To find the value of x, solve the equation 3x - 12 = 4x + 2:First, subtract 3x from both sides:-12 = x + 2.Then, subtract 2 from both sides:-14 = x.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Expressing bases as powers
When solving exponential equations, the first step is to express the bases as powers of a common number. This makes it easier to compare exponents and solve the equation. For example, in the equation \(27^{x-4}=9^{2x+1}\), notice that both 27 and 9 can be expressed as powers of 3: 27 is the same as \(3^3\) and 9 is the same as \(3^2\). By rewriting the equation with these expressions, we get: \[3^{3(x-4)} = 3^{2(2x+1)}\]. This transformation sets a common base, helping to make the problem simpler.
Simplifying exponents
The next step in solving the equation involves simplifying the exponents. This means we need to expand and combine like terms inside the exponents. Taking our previous equation \(3^{3(x-4)} = 3^{2(2x+1)}\), let鈥檚 simplify the exponents:
\[3^{3x - 12} = 3^{4x + 2}\]. Expand and multiply the coefficients inside the exponents:
  • In \(3(x-4)\), you get \[3x - 12\]
  • In \(2(2x+1)\), you get \[4x + 2\]
Now, the equation is simplified, making it easier to handle in the next steps.
Setting exponents equal
Since the bases are the same, we can set the exponents equal to each other and ignore the common base. This is a crucial step because it transforms an exponential equation into a linear one, which is simpler to solve. With our simplified equation \[3x - 12 = 4x + 2\], we can focus on just the exponents:
Set \[3x - 12 = 4x + 2\] and prepare to solve for x.
Solving linear equations
Now, solving the linear equation involves isolating the variable x. This typically includes moving terms from one side of the equation to the other and simplifying. Starting with
\[3x - 12 = 4x + 2\], follow these steps:
  • Subtract \(3x\) from both sides:
    \[ -12 = x + 2\]
  • Subtract 2 from both sides:
    \[ -14 = x\]
This leaves us with \[x = -14\]. By solving the linear equation, we鈥檝e found the value of x that satisfies the original exponential equation.

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Most popular questions from this chapter

A table of data is given. a. Graph the points and from visual inspection, select the model that would best fit the data. Choose from $$\begin{array}{ll} y=m x+b \text { (linear) } & y=a b^{x} \text { (exponential) } \\ y=a+b \ln x \text { (logarithmic) } & y=\frac{c}{1+a e^{-b x}} \text { (logistic) } \end{array}$$ b. Use a graphing utility to find a function that fits the data. $$ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 640 \\ \hline 20 & 530 \\ \hline 40 & 430 \\ \hline 50 & 360 \\ \hline 80 & 210 \\ \hline 100 & 90 \\ \hline \end{array} $$

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(21,000=63,000 e^{-0.2 t}\)

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Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(\log \left(p^{2}+6 p\right)=\log 7\)

Determine whether the two functions are inverses. \(h(x)=7 x-3\) and \(k(x)=\frac{x+3}{7}\)
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